Harmonic extensions of distributions

Abstract

We obtain harmonic extensions to the upper half-space of distributions in the weighted spaces w n+1 D L 1 , which according to [1] are the optimal spaces of tempered distributions S -convolvable with the classical euclidean version of the Poisson kernel . We also characterize the class of harmonic functions in the upper half-space with boundary * Partially supported by PAPIIT-IN105801. values in w n+1 D L 1 , generalizing in this way a classical result in the theory of Hardy spaces. Some facts concerning harmonic extensions of distributions in D L p , 1 < p ≤ ∞, are also approached in this paper, as well as natural relations among these spaces and the weighted spaces w n+1 D L 1 . We can also obtain n-harmonic extensions of appropriate weighted integrable distributions associated to a natural product domain version of the Poisson kernel.

N‐harmonic extensions of weighted integrable distributions

“…We conclude this section stating the following known technical lemma, whose detailed proof can be seen in [2].…”

“…P. Sjögren observes in [14] that the quantities R n w −n−1 (x)d × |μ|(x)+|a| and W −n−1 y −1 u 1,∞ in Theorem 14 are equivalent, that is to say they are either both zero or their ratio remains bounded from below and from above by positive constants. Using this observation, it can be proved as in the Euclidean case ( [2]) that there is a common bound on the number of terms, the order of the derivatives and the quasi-norms W D L 1 (R) for each j l , and c I belongs to a bounded subset of C. These common bounds and bounded sets can be described by constants that are equivalent quantities.…”

“…The next theorem characterizes those tempered distributions which are S -convolvable with the classical Poisson kernel P y defined in (2).…”

“…In fact, we have the following formula to compute the function T * P y (x): [2]). The intrinsic definition given in Definition 1 shows that the formula (4) is independent of the representation of the distribution T as in (1).…”

“…where T 1 ∈ E , T 2 ∈ D L 1 and T 2 is zero in a neighborhood of zero ( [2]). That this is the case follows quite readily from (22).…”